Existing tractable DSGE models at the lower bound have been celebrated for their analytical clarity and associated graphical representations. We show that expectations in these models are crucial, but conflict with Professional Forecaster’s data. We develop a new stochastic algorithm that nests these contributions, but can accommodate the necessary endogenous persistence to match expectations data. We show how to construct analytical solutions and graphical representations in that context. Using these, we study the government spending multiplier in a DSGE model with habit formation that matches U.S/Japanese expectations data. We find output multipliers that are below one in each case.